O teste da raz˜ao de verossimilhan¸cas ´e um procedimento de teste utilizado para comparar dois modelos que sejam encaixados. Por exemplo, no ajuste de um modelo de regress˜ao linear simples com uma ´unica vari´avel preditora, esse teste pode ser utilizado para testar o modelo nulo (somente o intercepto) contra o modelo com a vari´avel preditora. Nesse sentido, considere y = (y1, y2, . . . , yn)
um vetor de observa¸c˜oes da vari´avel aleat´oria Y , cujo modelo probabil´ıstico associado ´e f (y; θ) e θ∈ Θ ´e o vetor param´etrico de interesse. A estat´ıstica do teste da raz˜ao de verossimilhan¸cas pode ser escrita da seguinte forma
G2 = −2 log sup θ∈Θ0 L (θ|y) sup θ∈Θ L (θ|y) . (A.4)
Para grandes amostras mostra-se que, sob H0, a estat´ıstica G2 tem distribui¸c˜ao assint´otica Chi-
Quadrado com k − p graus de liberdade, sendo k o n´umero de parˆametros do modelo completo (espa¸co param´etrico irrestrito), e p o n´umero de parˆametros do modelo em teste, com Θ0 ⊂ Θ.
Na pr´atica, o teste da raz˜ao de verossimilhan¸cas ´e geralmente preferido para o teste de hip´oteses precisas, uma vez que, se existe um teste ϕ UMP, o lema de Neyman-Pearson o define. Em outras palavras, dentre todos os procedimentos para o teste de hip´oteses precisas, o teste da raz˜ao de verossimilhan¸cas ´e o uniformemente mais poderoso. Por outro lado, para testes de hip´oteses com- postas, podemos utilizar a extens˜ao do teste denominada raz˜ao de verossimilhan¸cas generalizada para um n´ıvel α previamente fixado.
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